2018 Grade 10 Math Challenge – National Level – Team Orals

Below are the 2018 Grade 10 Math Challenge – National Level – Team Orals questions with answers. More reviewers can be found on the Past Tests and All Posts pages.

Easy

1.) Find the reciprocal of the sum of the reciprocals of x - y and x+y . Write your answer as a single expression.

2.) A circle and a square have equal areas.The circle has circumference 2\pi. What is the length of the side of the square.

3.) How many 5-letter strings can be made from the word MANGO if each string should start with a vowel.

4.) A circle is divided by 3 points into 3 non-overlapping arcs whose measures are in the ratio 5 ∶ 5 ∶ 2. Find the smallest angle of the triangle formed by the 3 points.

5.) Two fair dice are rolled? What is the probability that the product of the resulting numbers is an odd prime?



6.) What is the area of the circle which passes through the points (0, 0), (5, 0), and (0, \sqrt{11})?

7.) Solve for x in the equation x^3 + 3x^2 + 3x = 100.

8.) A point is chosen on a number line so that the sum of its distances from the points 1, 2, 3, 4, 5, and 6, is as small as possible. What is the value of this sum?

9.) Find the sum of the infinite geometric series with first term 1, if the 3rd term is 27 times the 6th term.

10.) Solve for x in the equation \sqrt{x - 6} + \sqrt{x + 6} = 6.

Average

1.) Find the coordinates of the vertex of y = (x - 3)^2 + (x - 2)^2 + (x + 2)^2 + (x + 3)^2.

2.) A geometric sequence has 1st term 64 while the 7th and 11th terms are in the ratio 2 ∶ 3. Find the 17th term.

3.) Find the remainder when 5x^4 + 3x^3 - 9x^2 + 32x - 4 is divided by x + 3.

4.) An equilateral triangle and a regular hexagon have the same perimeters. What is the ratio of the area of the triangle to the area of the hexagon?

5.) Let q and r be the quotient and remainder, respectively, when a positive integer n is divided by 1000. For how many positive integers n having at most 5 digits is q + r divisible by 111?

Difficult

1.) Inside square ABCD, a point is chosen so that it is equidistant from 3 points: the vertices A and B, and the midpoint of side CD. If this common distance is 5 cm, how long (in cm) is a side of the square?

2.) A number is picked from the list of 5-digit positive integers whose digits have a sum of 43. Find the probability that this number has an even digit.

3.) Evaluate \dfrac{1}{x^2 - xy - xz - yz} + \dfrac{1}{z^2 - zx - zy + xy} when x = 1.8, y = 1.7 and z = 2.2.

4.) In a right triangle, the medians drawn from the vertices of the acute angles have lengths 9 and 13. Find the length of the hypotenuse.

5.) In an arithmetic sequence a_1, a_2, a_3, ... suppose a_5 + a_9 + a_13 = 5 and a_5 + a_6 + a_7 + a_8 + a_9 + a_{10} = 9. Find the index m such that a_m = {11}.

Answer Key

Easy

1.) \dfrac{x^2 - y^2}{2x}
2.) \sqrt{\pi}
3.) 48
4.) 30°
5.) 1/9
6.) 9\pi
7.) x = \sqrt[3]{101} - 1
8.) 7
9.) 3/2
10.) 10

Average

1.) (0, 26)
2.) 324
3.) 143
4.) 2 : 3
5.) 900

Difficult

1.) 8 cm
2.) 2/3
3.) -20
4.) 10\sqrt{2}
5.) 93

View Team Orals here: 2018 Grade 6 Math Challenge – National Level – Individual Finals

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